Question

Suppose that people arrive at a bus stop in accordance with a Poisson process with rate \(\lambda\). The bus departs at time \(t\). Let \(X\) denote the total amount of waiting time of all those who get on the bus at time \(t\). We want to determine \(\operatorname{Var}(X)\). Let \(N(t)\) denote the number of arrivals by time \(t\). (a) What is \(E[X \mid N(t)] ?\) (b) Argue that \(\operatorname{Var}[X \mid N(t)]=N(t) t^{2} / 12\) (c) What is \(\operatorname{Var}(X) ?\)

Short answer

The variance of the total waiting time, Var(X), is given by \(\frac{\lambda t^{3}}{3}\).

Step-by-step solution

(a) Finding E[X | N(t)]

Let's find the expected value of X given N(t), the number of arrivals by time t. The waiting time for each person is uniformly distributed between 0 and t. Therefore, we have: E[X | N(t)] = N(t) * E[waiting time for each person] = N(t) * (t / 2), since the expected value of a uniform distribution over the range [0, t] is t/2.

(b) Finding Var[X | N(t)]

Now, let's find the conditional variance Var[X | N(t)]. Since the waiting times are independent and uniformly distributed, we consider the variance of the waiting time for each person and sum it up N(t) times: Var[X | N(t)] = N(t) * Var[waiting time for each person] = N(t) * (t^2 / 12), since the variance of a uniform distribution over the range [0, t] is t^2/12.

(c) Finding Var(X)

Now, we can find the variance of X by using the conditional variance that we found in step (b) and the conditional expectation that we found in step (a) by applying the law of total variance: Var(X) = E[Var[X | N(t)]] + Var(E[X | N(t)]) First, we find E[Var[X | N(t)]]: E[Var[X | N(t)]] = E[N(t) * (t^2 / 12)] = E[N(t)] * (t^2 / 12) = (λt) * (t^2 / 12), since E[N(t)] = λt as N(t) follows a Poisson distribution with rate λ. Next, we find Var(E[X | N(t)]): Var(E[X | N(t)]) = Var(N(t) * (t / 2)) = (t / 2)^2 * Var(N(t)) = (t / 2)^2 * (λt), since Var(N(t)) = λt as N(t) follows a Poisson distribution with rate λ. Finally, substituting the results from above: Var(X) = (λt) * (t^2 / 12) + (t / 2)^2 * (λt) = (λt^3/12) + (λt^3/4) = (λt^3/3). Therefore, the variance of the total waiting time, Var(X), is given by \(\frac{\lambda t^{3}}{3}\).

Key concepts

Conditional Expectation

The concept of conditional expectation is pivotal in understanding complex random processes. It describes the average outcome of a random variable given that certain conditions are met. In the case of the bus stop problem, the conditional expectation, denoted as E[X | N(t)], represents the expected total waiting time of all people at the bus stop given the number of arrivals by a certain time.
Taking into consideration that the waiting time for each individual is uniformly distributed between 0 to t, each person's average waiting time is simply the midpoint of that interval, which is t/2. Therefore, if N(t) people arrive by time t, the total expected waiting time will be N(t) multiplied by t/2. This method of calculation helps ensure that predictions about the system's behavior are accurate and reflective of all possible outcomes.

Conditional Variance

While conditional expectation gives us an average, conditional variance measures the spread or variability around that average, under certain conditions. In our Poisson process example, once we know the number of arrivals N(t), we seek to determine how much the total waiting time X can fluctuate around its expected value.
For independent and uniform waiting times, the conditional variance is calculated by adding the variances of individual waiting times N(t) times. Since the variance for a uniform distribution between 0 and t (t^2/12) reflects the spread of an individual's waiting time, the total spread or conditional variance for N(t) arrivals is N(t) times t^2/12. This offers a quantifiable way to gauge uncertainty in the system, an essential consideration for predicting and managing waiting times at the bus stop.

Uniform Distribution

In statistics, a uniform distribution is a type of probability distribution in which all outcomes are equally likely. The waiting times at the bus stop are an example of a continuous uniform distribution, as they can take any value between 0 and t with equal likelihood. Two key characteristics of this distribution used in our problem are the mean (t/2) and the variance (t^2/12).

Properties of Uniform Distribution

• Mean: The average value is the midpoint of the range, making calculations straightforward.
• Variance: Given by the formula (end point - start point)^2/12 which quantifies the extent to which individual data points spread out from the mean.

These characteristics are instrumental in establishing the building blocks for more complex calculations in our problem, like finding the total expected waiting time and its variability.

Law of Total Variance

The law of total variance breaks down the total variance of a random variable into two parts: the expected value of the conditional variances and the variance of the conditional means. This law is represented by the equation Var(X) = E[Var[X | Y]] + Var(E[X | Y]), where X is the random variable of interest and Y is another random variable upon which X is conditioned.
In the bus stop scenario, we apply this law by first calculating the expected conditional variance, which incorporates the variability of individual wait times, and then computing the variance of the conditional expectations, representing the variability in the number of arrivals. These calculations allow us to integrate both sources of uncertainty to find the total variance of the waiting time, offering a complete picture of the variability in the system.

Random Arrival

The notion of random arrival is central to models like the Poisson process, which describes events occurring randomly over time. In our exercise, the arrivals at the bus stop by time t is a textbook case of a Poisson process. By designating arrivals as random, we acknowledge that they are independent of each other and the probability of an arrival in any given interval is the same.
Understanding the randomness of arrivals is essential for using statistical techniques like the Poisson distribution to predict how many individuals will arrive in a set period and informs the use of conditional expectations and variances. It ensures we have an adaptable and robust methodology for quantifying and managing the intrinsic randomness encountered in real-world scenarios, such as bus stop arrivals.